A linearly similar Sz.-Nagy-Foiaş model in a domain. (English. Russian original) Zbl 1067.47014

Yakubovich, D. V.

St. Petersbg. Math. J. 15, No. 2, 289-321 (2004); translation from Algebra Anal. 15, No. 2, 190-237 (2003).

The paper deals with a linearly similar functional model for linear operators. Let \(\Omega_{\text{int}}\) and \(\Omega_{\text{ext}}\) be open disjoint sets in \({\mathbb C}\) such that \({\mathbb C}=\Omega_{\text{int}}\cup\Gamma\cup\Omega_{\text{ext}}\), where \(\Gamma\) is a union of finitely many piecewise smooth contours, and let \(\delta\) be an operator valued \(H^\infty(\Omega_{\text{int}})\) function such that \(\delta^{-1}\) is meromorphic on \(\Omega_{\text{int}}\) and uniformly bounded a.e. on \(\Gamma\). In the scalar case, the model space \({\mathcal H}(\delta)\) is defined as a Hilbert space of functions \(f\) from the Smirnov class \(E^2(\Omega_{\text{ext}})\) which are meromorphic on \(\Omega_{\text{int}}\) and satisfy \(\delta f| _{\Omega_{\text{int}}}\in E^2(\Omega_{\text{int}})\), \(f_{\text{int}}=f_{\text{ext}}\) a.e. on \(\Gamma\).
The author investigates the operators which are linearly similar to the operator of multiplication by \(z\) in some quotient space \(E^2(\Omega_{\text{int}})/\delta E^2(\Omega_{\text{int}})\). In this case, the function \(\delta\) is called a generalized characteristic function of \(A\). If \(\Omega_{\text{int}}\) is a simply connected domain and \(\delta\) is a \(*\)-inner function, then this model is related to the Sz.-Nagy-Foias model. In the case of a multiply connected domain, generalizations of the Sz.-Nagy-Foias model were considered by {J. A. Ball [J. Oper. Theory 1, 3–25 (1979; Zbl 0435.47035)], S. McCullough [Integral Equations Oper. Theory 35, No. 1, 65–84 (1999; Zbl 0935.47007)], and by B. Pavlov [St. Petersbg. Math. J. 6, No. 3, 619–633 (1995; Zbl 0839.47007)].
As examples, the author considers \(C_{0\cdot}\) contractions and dissipative operators in Hilbert spaces, differentiation with nondissipative boundary conditions, generators of systems with delay, and calculates the corresponding characteristic functions. The relationship with the exact controllability of linear systems is studied. An important feature of the theory is the duality of models. The reproducing kernel method is applied in order to calculate the corresponding characteristic function \(\delta\) via duality. The problem of nonuniqueness of linearly similar models is also discussed.}

Reviewer: Vladimir A. Derkach (Donetsk)

MSC:

47A45	Canonical models for contractions and nonselfadjoint linear operators
47A20	Dilations, extensions, compressions of linear operators
47A48	Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.

Keywords:

contraction; dissipative operator; generalized characteristic function; duality of models; reproducing kernel method; nonuniqueness of similar models

Citations:

Zbl 0435.47035; Zbl 0935.47007; Zbl 0839.47007

References:

[1]	D. Z. Arov, Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), no. 2, 211 – 228, 457 (Russian). · Zbl 0414.93014
[2]	D. Z. Arov and M. A. Nudel\(^{\prime}\)man, Conditions for the similarity of all minimal passive realizations of a given transfer function (scattering and resistance matrices), Mat. Sb. 193 (2002), no. 6, 3 – 24 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 791 – 810. · Zbl 1085.47016 · doi:10.1070/SM2002v193n06ABEH000657
[3]	V. Vasyunin and S. Kupin, Criteria for the similarity of a dissipative integral operator to a normal operator, Algebra i Analiz 13 (2001), no. 3, 65 – 104 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 389 – 416. · Zbl 1032.47019
[4]	I. C. Gohberg and M. G. Kreĭn, On a description of contraction operators similar to unitary ones, Funkcional. Anal. i Priložen. 1 (1967), 38 – 60 (Russian).
[5]	G. M. Gubreev, Spectral theory of regular quasi-exponentials and regular \?-representable vector functions (the projection method: 20 years later), Algebra i Analiz 12 (2000), no. 6, 1 – 97 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 875 – 947.
[6]	V. V. Kapustin, Spectral analysis of almost unitary operators, Algebra i Analiz 13 (2001), no. 5, 44 – 68 (Russian); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 739 – 756.
[7]	A. L. Lihtarnikov and V. A. Jakubovič, A frequency theorem for equations of evolution type, Sibirsk. Mat. Ž. 17 (1976), no. 5, 1069 – 1085, 1198 (Russian). · Zbl 0375.93031
[8]	M. M. Malamud, A criterion for a closed operator to be similar to a selfadjoint operator, Ukrain. Mat. Zh. 37 (1985), no. 1, 49 – 56, 134 (Russian).
[9]	M. M. Malamud, On the similarity of a triangular operator to a diagonal operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 28, 201 – 241, 367 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 115 (2003), no. 2, 2199 – 2222. · Zbl 1025.47012 · doi:10.1023/A:1022888921572
[10]	S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86 – 114, 203 (Russian). Boundary value problems of mathematical physics, 10. · Zbl 0445.47010
[11]	S. N. Naboko, Conditions for similarity to unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 16 – 27 (Russian).
[12]	S. N. Naboko and M. M. Faddeev, Operators of the Friedrichs model that are similar to a selfadjoint operator, Vestnik Leningrad. Univ. Fiz. Khim. vyp. 4 (1990), 78 – 82, 114 (Russian, with English summary). · Zbl 1268.47023
[13]	Лекции об операторе сдвига, ”Наука”, Мосцощ, 1980 (Руссиан). Н. К. Никол\(^{\приме}\)ский, Треатисе он тхе шифт оператор, Грундлехрен дер Матхематисчен Щиссенсчафтен [Фундаментал Принциплес оф Матхематицал Сциенцес], вол. 273, Спрингер-Верлаг, Берлин, 1986. Спецтрал фунцтион тхеоры; Щитх ан аппендиш бы С. В. Хруščев [С. В. Хрущёв] анд В. В. Пеллер; Транслатед фром тхе Руссиан бы Јаак Пеетре.
[14]	B. S. Pavlov, Conditions for separation of the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 123 – 148, 240 (Russian).
[15]	B. Pavlov, Nonphysical sheet for perturbed Jacobian matrices, Algebra i Analiz 6 (1994), no. 3, 185 – 199; English transl., St. Petersburg Math. J. 6 (1995), no. 3, 619 – 633. · Zbl 0829.47011
[16]	I. I. Privalov, Boundary properties of analytic functions, GITTL, Moscow-Leningrad, 1950. (Russian)
[17]	V. A. Ryzhov, Absolutely continuous and singular subspaces of a nonselfadjoint operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23, 163 – 202, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 5, 3886 – 3911. · Zbl 0909.47004 · doi:10.1007/BF02355830
[18]	L. A. Sahnovič, Dissipative operators with absolutely continuous spectrum, Trudy Moskov. Mat. Obšč. 19 (1968), 211 – 270 (Russian).
[19]	B. M. Solomyak, A functional model for dissipative operators. A coordinate-free approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 18, 57 – 91, 184 – 185 (Russian, with English summary); English transl., J. Soviet Math. 61 (1992), no. 2, 1981 – 2002. · Zbl 0784.47018 · doi:10.1007/BF01095663
[20]	Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. · Zbl 0201.45003
[21]	S. I. Fedorov, On harmonic analysis in a multiply connected domain and character-automorphic Hardy spaces, Algebra i Analiz 9 (1997), no. 2, 192 – 240 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 339 – 378. · Zbl 0892.32004
[22]	A. V. Štraus, Characteristic functions of linear operators, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 43 – 74 (Russian).
[23]	E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. · Zbl 0078.10004
[24]	V. A. Jakubovič, The frequency theorem for the case in which the state space and the control space are Hilbert spaces, and its application in certain problems in the synthesis of optimal control. II, Sibirsk. Mat. Ž. 16 (1975), no. 5, 1081 – 1102, 1132 (Russian).
[25]	Daniel Alpay and Victor Vinnikov, Finite dimensional de Branges spaces on Riemann surfaces, J. Funct. Anal. 189 (2002), no. 2, 283 – 324. · Zbl 1038.46020 · doi:10.1006/jfan.2000.3623
[26]	D. Z. Arov and M. A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), no. 1, 1 – 45. · Zbl 0838.47004 · doi:10.1007/BF01195483
[27]	Z. D. Arova, On \?-unitary nodes with strongly regular \?-inner characteristic functions in the Hardy class \?^{\?\times \?}\(_{2}\), Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 29 – 38. · Zbl 1020.47006
[28]	Sergei A. Avdonin and Sergei A. Ivanov, Families of exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems; Translated from the Russian and revised by the authors. · Zbl 0866.93001
[29]	Joseph A. Ball, A lifting theorem for operator models of finite rank on multiply-connected domains, J. Operator Theory 1 (1979), no. 1, 3 – 25. · Zbl 0435.47035
[30]	Joseph A. Ball and Nir Cohen, de Branges-Rovnyak operator models and systems theory: a survey, Topics in matrix and operator theory (Rotterdam, 1989) Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 93 – 136. · Zbl 0756.47007 · doi:10.1007/978-3-0348-5672-0_5
[31]	Nour-Eddine Benamara and Nikolai Nikolski, Resolvent tests for similarity to a normal operator, Proc. London Math. Soc. (3) 78 (1999), no. 3, 585 – 626. · Zbl 1028.47500 · doi:10.1112/S0024611599001756
[32]	Louis de Branges and James Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965) Wiley, New York, 1966, pp. 295 – 392.
[33]	Douglas N. Clark, On a similarity theory for rational Toeplitz operators, J. Reine Angew. Math. 320 (1980), 6 – 31. · Zbl 0441.47038 · doi:10.1515/crll.1980.320.6
[34]	Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. · Zbl 0839.93001
[35]	R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610-616. · Zbl 0211.16802
[36]	Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
[37]	Harry Dym, On Riccati equations and reproducing kernel spaces, Recent advances in operator theory (Groningen, 1998) Oper. Theory Adv. Appl., vol. 124, Birkhäuser, Basel, 2001, pp. 189 – 215. · Zbl 0989.46015
[38]	Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. · Zbl 0718.47010
[39]	Paul A. Fuhrmann, Linear systems and operators in Hilbert space, McGraw-Hill International Book Co., New York, 1981. · Zbl 0456.47001
[40]	Paul A. Fuhrmann, A polynomial approach to linear algebra, Universitext, Springer-Verlag, New York, 1996. · Zbl 0852.15001
[41]	Morisuke Hasumi, Invariant subspace theorems for finite Riemann surfaces, Canad. J. Math. 18 (1966), 240 – 255. · Zbl 0172.41603 · doi:10.4153/CJM-1966-027-1
[42]	Birgit Jacob and Jonathan R. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups, Integral Equations Operator Theory 40 (2001), no. 2, 231 – 243. · Zbl 1031.93107 · doi:10.1007/BF01301467
[43]	B. Jacob, J. R. Partington, and S. Pott, Conditions for admissibility of observation operators and boundedness of Hankel operators (to appear). · Zbl 1046.47036
[44]	M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc. 334 (1992), no. 2, 479 – 517. · Zbl 0768.34041
[45]	N. J. Kalton and C. Le Merdy, Solution of a problem of Peller concerning similarity, J. Operator Theory 47 (2002), no. 2, 379 – 387. · Zbl 1019.47023
[46]	V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. · Zbl 0937.93003
[47]	Stanislav Kupin, Linear resolvent growth test for similarity of a weak contraction to a normal operator, Ark. Mat. 39 (2001), no. 1, 95 – 119. · Zbl 1031.47003 · doi:10.1007/BF02388793
[48]	S. Kupin and S. Treil, Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator, Illinois J. Math. 45 (2001), no. 1, 229 – 242. · Zbl 0994.47006
[49]	Christian Le Merdy, On dilation theory for \?\(_{0}\)-semigroups on Hilbert space, Indiana Univ. Math. J. 45 (1996), no. 4, 945 – 959. · Zbl 0876.47009 · doi:10.1512/iumj.1996.45.1191
[50]	Christian Le Merdy, The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum 56 (1998), no. 2, 205 – 224. · Zbl 0998.47028 · doi:10.1007/PL00005942
[51]	-, The Weiss conjecture for bounded analytic semigroups (to appear). · Zbl 1064.47045
[52]	Christian Le Merdy, A bounded compact semigroup on Hilbert space not similar to a contraction one, Semigroups of operators: theory and applications (Newport Beach, CA, 1998) Progr. Nonlinear Differential Equations Appl., vol. 42, Birkhäuser, Basel, 2000, pp. 213 – 216. · Zbl 0982.47030
[53]	-, Similarities of \(\omega \)-accretive operators, Prépubl. Lab. Math. Besançon no. 2002/07, Univ. France Comté, Centre Nat. Rech. Sci., Besançon, 2002.
[54]	J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations].
[55]	M. S. Livšic, N. Kravitsky, A. S. Markus, and V. Vinnikov, Theory of commuting nonselfadjoint operators, Mathematics and its Applications, vol. 332, Kluwer Academic Publishers Group, Dordrecht, 1995. · Zbl 0834.47004
[56]	Sjoerd M. Verduyn Lunel and Dmitry V. Yakubovich, A functional model approach to linear neutral functional-differential equations, Integral Equations Operator Theory 27 (1997), no. 3, 347 – 378. · Zbl 0891.34077 · doi:10.1007/BF01324734
[57]	Alan McIntosh, Operators which have an \?_{\infty } functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210 – 231. · Zbl 0634.47016
[58]	Scott McCullough, Commutant lifting on a two holed domain, Integral Equations Operator Theory 35 (1999), no. 1, 65 – 84. · Zbl 0935.47007 · doi:10.1007/BF01225528
[59]	N. K. Nikolski, Contrôlabilité à une renormalisation près et petits opérateurs de contrôle, J. Math. Pures Appl. (9) 77 (1998), no. 5, 439 – 479 (French, with English summary). · Zbl 0910.93012 · doi:10.1016/S0021-7824(98)80027-5
[60]	Nikolai Nikolski and Vasily Vasyunin, Elements of spectral theory in terms of the free function model. I. Basic constructions, Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 211 – 302. · Zbl 0999.47010
[61]	Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author.
[62]	Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. · Zbl 0614.47006
[63]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[64]	Gilles Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), no. 2, 351 – 369. · Zbl 0869.47014
[65]	A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite-dimensional systems with unbounded input and output operators, SIAM J. Control Optim. 25 (1987), no. 1, 121 – 144. · Zbl 0615.93039 · doi:10.1137/0325009
[66]	Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim. 37 (1999), no. 1, 131 – 164. · Zbl 0955.49018 · doi:10.1137/S0363012996314257
[67]	Olof Staffans and George Weiss, Transfer functions of regular linear systems. II. The system operator and the Lax-Phillips semigroup, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3229 – 3262. · Zbl 0996.93012
[68]	N. G. Makarov and V. I. Vasjunin, A model for noncontractions and stability of the continuous spectrum, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 365 – 412. · Zbl 0461.47003
[69]	Michael Voichick, Invariant subspaces on Riemann surfaces, Canad. J. Math. 18 (1966), 399 – 403. · Zbl 0147.11501 · doi:10.4153/CJM-1966-042-8
[70]	George Weiss, Admissible observation operators for linear semigroups, Israel J. Math. 65 (1989), no. 1, 17 – 43. · Zbl 0696.47040 · doi:10.1007/BF02788172
[71]	George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim. 27 (1989), no. 3, 527 – 545. · Zbl 0685.93043 · doi:10.1137/0327028
[72]	Daoxing Xia, On the kernels associated with a class of hyponormal operators, Integral Equations Operator Theory 6 (1983), no. 3, 444 – 452. · Zbl 0554.47009 · doi:10.1007/BF01691907
[73]	D. V. Yakubovich, Spectral properties of smooth finite rank perturbations of a planar Lebesgue spectrum cyclic normal operator, Report no. 15 (1990/91), Inst. Mittag-Leffler, Stockholm.
[74]	Dmitry V. Yakubovich, Spectral properties of smooth perturbations of normal operators with planar Lebesgue spectrum, Indiana Univ. Math. J. 42 (1993), no. 1, 55 – 83. · Zbl 0799.47003 · doi:10.1512/iumj.1993.42.42005
[75]	D. V. Yakubovich, Dual piecewise analytic bundle shift models of linear operators, J. Funct. Anal. 136 (1996), no. 2, 294 – 330. · Zbl 0867.47010 · doi:10.1006/jfan.1996.0032
[76]	-, A similarity version of the Nagy-Foias model, duality and exact controllability, 17th International Conference on Operator Theory (Timisoara, 1998): Abstracts of Reports.
[77]	Dmitry V. Yakubovich, Subnormal operators of finite type. II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998), no. 3, 623 – 681. · Zbl 0933.47016 · doi:10.4171/RMI/247
[78]	Enrique Zuazua, Some problems and results on the controllability of partial differential equations, European Congress of Mathematics, Vol. II (Budapest, 1996) Progr. Math., vol. 169, Birkhäuser, Basel, 1998, pp. 276 – 311. · Zbl 1126.35305
[79]	A. S. Tikhonov, A functional model and duality of spectral components for operators with a continuous spectrum on a curve, Algebra i Analiz 14 (2002), no. 4, 158 – 195 (Russian); English transl., St. Petersburg Math. J. 14 (2003), no. 4, 655 – 682. · Zbl 1052.47006
[80]	Piotr Grabowski and Frank M. Callier, Admissible observation operators. Semigroup criteria of admissibility, Integral Equations Operator Theory 25 (1996), no. 2, 182 – 198. · Zbl 0856.93021 · doi:10.1007/BF01308629